Fluctuating Navier-Stokes equations for inelastic hard spheres or disks
Abstract
Starting from the fluctuating Boltzmann equation for smooth inelastic hard spheres or disks, closed equations for the fluctuating hydrodynamic fields to Navier-Stokes order are derived. This requires deriving constitutive relations for both the fluctuating fluxes and the correlations of the random forces. The former are identified as having the same form as the macroscopic average fluxes and involving the same transport coefficients. On the other hand, the random force terms exhibit two peculiarities as compared with their elastic limit for molecular systems. First, they are not white but have some finite relaxation time. Second, their amplitude is not determined by the macroscopic transport coefficients but involves new coefficients.
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PHYSICAL REVIEW E 83, 041303 (2011) Fluctuating Navier-Stokes equations for inelastic hard spheres or disks J. Javier Brey, P. Maynar, and M. I. Garc´ ıa de Soria F´ ısica Te´ orica, Universidad de Sevilla, Apartado de Correos 1065, E-41080 Sevilla, Spain (Received 29 December 2010; published 20 April 2011) Starting from the fluctuating Boltzmann equation for smooth inelastic hard spheres or disks, closed equations for the fluctuating hydrodynamic fields to Navier-Stokes order are derived. This requires deriving constitutive relations for both the fluctuating fluxes and the correlations of the random forces. The former are identified as having the same form as the macroscopic average fluxes and involving the same transport coefficients. On the other hand, the random force terms exhibit two peculiarities as compared with their elastic limit for molecular systems. First, they are not white but have some finite relaxation time. Second, their amplitude is not determined by the macroscopic transport coefficients but involves new coefficients. DOI: 10.1103/PhysRevE.83.041303 PACS number(s): 45.70.−n, 05.20.Dd, 05.60.−k, 51.10.+y I. INTRODUCTION Granular matter is ubiquitous in nature. It has a tremendous impact on a wide range of industries and also raises important conceptual challenges. When the grains move freely and independently between collisions the system is referred to as a granular gas [1]. The simplest model of a granular gas at the particle level of description is an ensemble of inelastic hard spheres or disks [2]. In the past decade or so, a great deal of progress and understanding of granular gases has been achieved [3,4] by using this model, especially in the dilute limit in which the average behavior of the gas is accurately described by the inelastic Boltzmann equation [5,6]. This includes derivation of the hydrodynamic Navier-Stokes equations with explicit expressions for the transport coefficients, study of several Newtonian and non-Newtonian steady states, analysis of the spectrum of the linearized inelastic Boltzmann operator, identification of several instabilities both in freely evolving and heated granular gases, and detailed characterization of the distribution function of several relevant states [7]. The theoretical predictions have been corroborated by molecular dynamics and direct Monte Carlo simulation results. On the other hand, knowledge about fluctuations and correlations in dilute granular gases is much more limited, although they are known to play a fundamental role in the macroscopic behavior of granular flows in many cases. The purpose of this paper is to derive Langevin-like equations describing the dynamics of the fluctuating hydrodynamic fields in a freely evolving homogenous dilute granular gas, namely in the usually termed homogeneous cooling state (HCS) [8]. Of course, in the elastic limit the equations reduce to the fluctuating hydrodynamic equations proposed by Landau and Lifshitz for equilibrium molecular fluids [9]. The starting point will be the fluctuating Langevin-Boltzmann equation derived in Ref. [10] by using nonequilibrium statistical mechanics methods. The equation has the same mathematical form as the linearized inelastic Boltzmann equation plus and additive white-noise term. Moreover, it is derived under the same conditions as required for the nonlinear Boltzmann equation, namely the low-density limit. Hydrodynamic fluctuations in freely evolving granular gases have been already partially investigated. First studies used mesoscopic equations in which the Landau and Lifschitz fluctuating hydrodynamic equations for molecular gases were directly modified to incorporate the effect of energy dissipation in collisions, the resulting equations being expected to be valid in the quasielastic limit [11,12]. In particular, the correlations of the noise terms were assumed to be proportional to δtime functions (i.e., white noises) and determined by the NavierStokes transport coefficients, using the same expressions as for molecular systems. Also, some results based on a single relaxation model kinetic equation have been reported [13]. In the above studies the interest focused on the initial buildup of spatial correlations of the hydrodynamic fields near the so-called clustering instability. Approaches starting from a more fundamental description of the granular gas, i.e., the Liouville equation of the system, have also been carried out. In particular, fluctuations and correlations of the total energy of an isolated granular gas have been computed and a good agreement between theory and simulations has been found [14,15]. Moreover, in Ref. [10], the fluctuations and correlations of the transversal component of the velocity field were studied in detail. It was shown that the Langevin equation obeyed by this field in a granular gas has two crucial differences with the elastic limit. First, the noise is not white but has a finite correlation time and, second, the amplitude of its second moment is not determined by the shear viscosity but by some new coefficient. It must be noted that the transversal velocity field is special in the sense that it evolves uncoupled from the other hydrodynamic fields. The purpose of this paper is to extend the analysis in Ref. [10] by deriving the whole closed set of fluctuating hydrodynamic equations for a dilute granular gas in the HCS. In this context, it must be realized that this state is the homogeneous reference state for granular gases, playing a role somehow similar to the equilibrium state in molecular fluids. Therefore, understanding fluctuations and correlations in this state is an unavoidable first step toward a deep knowledge of them in more complex and also more realistic states of granular gases. It is worth to mention also some previous studies of fluctuations dealing with driven granular gas models, in which the grains are assumed to be in contact with some external energy source or thermal bath [16–18]. Since the latter modifies the stochastic properties of the granular gas in a nontrivial way, it is not evident that there is a direct relation between fluctuations in driven models and the free model being considered here. 041303-1 1539-3755/2011/83(4)/041303(13) ©2011 American Physical Society
BREY, MAYNAR, AND GARC´ IA DE SORIA PHYSICAL REVIEW E 83, 041303 (2011) The plan of the remaining of the paper is as follows. In Sec. II, the Boltzmann-Langevin equation for smooth inelastic hard spheres or disks is shortly reviewed, and the appropriate dimensionless time and length scales are introduced. From the above kinetic equation, balance equations for the fluctuating hydrodynamic fields in the HCS are directly derived, as shown in Sec. III. These equations are worthless until they are closed by deriving constitutive relations for the fluxes appearing in them and also explicit expressions for the correlations of the noise terms. This is done here by introducing a projection operator over the hydrodynamic subspace of the distribution functions. This subspace is generated by the hydrodynamic eigenfunctions of the linear inelastic Boltzmann operator, whose definitions and expressions are also briefly noted. The equation for the number density involves neither flux nor noise term. The Langevin equation for the velocity field is derived in Sec. IV and the one for the energy field in Sec. V.In both cases, expressions for the fluctuating fluxes and for the correlation functions of the fluctuating forces are provided. These expressions are not exact but have been obtained under well-defined and controlled approximations. It is shown that the fluctuating forces have finite relaxation times which can be related with nonhydrodynamic eigenvalues of the linearized Boltzmann operator. For the sake of clarity, many details of the calculations as well as technical points are given in six appendixes. Finally, Sec. VI contains a short summary and some final remarks. II. BOLTZMANN-LANGEVIN EQUATION FOR SMOOTH INELASTIC HARD SPHERES OR DISKS The system being considered is a dilute gas of smooth inelastic hard spheres (d=3) or disks (d=2) of mass mand diameter σ. Inelasticity of collisions is modeled by means of a constant, velocity-independent, coefficient of normal restitution α, defined in the interval 0 <α⩽1. At a mesoscopic level, the system is described by a fluctuating or stochastic one-particle distribution function, F(r,v,t), whose average, denoted by f(r,v,t), is the usual one-particle distribution function giving the average number of particles density being at position rwith velocity vat time t. This average distribution obeys the nonlinear inelastic Boltzmann equation [5]. The system is assumed to be in the homogeneous cooling state (HCS), macroscopically characterized by a uniform number of particles density n, a vanishing flow velocity, and a uniform granular temperature TH(t) that decreases monotonically in time due to the energy dissipation in collisions. To describe fluctuations about the HCS, it is convenient to introduce dimensionless length and time scales defined by ≡ r λ(1) and s≡t 0 dt1 v0(t1) λ,(2) respectively. In the above expressions, λ≡(nσd−1)−1and v0(t)=2TH(t) m1/2 .(3) Then λis proportional to the mean free path of the gas in the HCS and v0(t) is a characteristic thermal velocity. The granular temperature is defined from the average kinetic energy of the grains with the Boltzmann constant formally set equal to unity. It is easily verified that the time scale sis proportional to the accumulated average number of collisions per particle occurring in the system in the time interval between 0 and t. Consistently with Eqs. (1) and (2), a new velocity scale is introduced through c≡v v0(t).(4) Also, the deviation of F(r,v,t) from its average in the HCS, fH(v,t), δF(r,v,t)≡F(r,v,t)−fH(v,t),(5) is transformed to a dimensionless form by defining δ F(,c,s)≡n−1vd 0(t)δF(r,v,t).(6) This function obeys the Boltzmann-Langevin equation [10] ∂ ∂s +c·∂ ∂−(c)δ F(,c,s)= S(,c,s),(7) where (c) is the linearized inelastic Boltzmann operator given in Appendix Aand the term S(,c,s) has the properties of a white-noise term, S(,c,s)H=0,(8) S(,c,s)δ F(,c,s)H=0,(9) for s>s and S(,c,s) S(,c,s)H=n−1λ−dδ(−)δ(s−s) (c,c). (10) Here and in the following, angular brackets with the subindex Hare used to denote stochastic average in the HCS. The amplitude (c,c) of the noise is given by (c,c)=−[(c)+(c)]χ(c)δ(c−c) +T0(c,c)χ(c)χ(c),(11) with T0(c,c) being the inelastic binary collision operator given in Eq. (A1) and χ(c) the dimensionless scaled velocity distribution of the HCS defined as χ(c)≡n−1vd 0(t)fH(v,t).(12) An accurate expression for this distribution is known and is reiterated in Appendix A[Eq. (A6)]. For the present purposes, it is useful to employ the Fourier representation and to introduce δ F(k,c,s)≡de−ik·δ F(,c,s).(13) In Fourier space, Eq. (7) becomes ∂ ∂s −(k,c)δ F(k,c,s)= S(k,c,s),(14) where (k,c)≡(c)−ik·c(15) 041303-2
FLUCTUATING NAVIER-STOKES EQUATIONS FOR ... PHYSICAL REVIEW E 83, 041303 (2011) is the linear inhomogeneous inelastic Boltzmann operator. The property of the noise in Eq. (10) now reads S(k,c,s) S(k,c,s)H= V2 Nδk,−kδ(s−s) (c,c),(16) δk,−kbeing the Kronecker δsymbol and V≡Vλ −dthe volume of the system in the length scale defined by λ. III. BALANCE EQUATIONS FOR THE FLUCTUATING HYDRODYNAMIC FIELDS Consider the mesoscopic number of particles density N(r,t), momentum density G(r,t), and energy density E(r,t). Dimensionless deviations from their average values in the HCS are introduced through the definitions δρ(k,s)≡N(k,s)−n n=dcδ F(k,c,s),(17) δω(k,s)≡δG(k,s) mnv0(t)=dccδ F(k,c,s),(18) δ(k,s)≡2 dnTH(t)E(k,s)−d 2nTH(s) =2 ddcc2δ F(k,c,s).(19) In Ref. [10], balance equations for these fields were derived by taking velocity moments in the fluctuating Boltzmann equation. In Fourier space they read ∂ ∂s δρ(k,s)+ik·δω(k,s)=0,(20) ∂ ∂s −ζ0 2δω(k,s)+ik·δ(k,s)=0,(21) ∂ ∂s δ(k,s)+id+2 d k·δω(k,s)+δζ(k,s) +i2 d k·δφ(k,s)= S(k,s).(22) In the above equations, ζ0is the cooling rate of the HCS. Its definition is given in Appendix A, where an approximated expression is also provided. Moreover, δ(k,s) and δφ(k,s) are the fluctuating pressure tensor and heat flux, respectively. They are functionals of the fluctuating distribution function, δ(k,s)≡δ(k,s) 2 I+dc(c)δ F(k,c,s),(23) δφ(k,s)=dc(c)δ F(k,c,s),(24) where Iis the unit tensor of dimension dand (c)≡cc −c2 d I,(25) (c)≡c2−d+2 2c.(26) The equation for the fluctuating energy field, Eq. (22), involves two terms vanishing in the elastic limit α→1. One is associate with the cooling rate fluctuations δζ(k,s)=−2 ddcc2(c)δ F(k,c,s),(27) and the other one is the noise term S(k,s) arising directly from the fluctuations in phase space, S(k,s)=2 ddcc2 S(k,c,s).(28) From Eqs. (8) and (16) it follows that S(k,s)H=0 (29) and S(k,s) S(k,s)H=4 V2 d2Nδk,−kδ(s−s) ×dcdcc2c2 (c,c).(30) Also it is S(k,s) F(k,c,s)H=0 (31) and S(k,s)δρ(k,s)H= S(k,s)δω(k,s)H = S(k,s)δ(k,s)H=0,(32) for s>s . Equations (20)–(22) are not closed, since they contain the quantities δ,δφ, and δζ, defined above in terms of δ F(k,c,s), as well as the noise term S(k,s). The aim in the following will be to obtain a self-consistent description for the fluctuating fields. In this context, it will be useful to consider the eigenvalue problem for the inelastic homogeneous linear Boltzmann operator [19,20], (c)ξβ(c)=λβξβ(c).(33) The solutions of this equation corresponding to the infinite wave length limit (k=0) of the hydrodynamic equations are given by [19,20] λ1=0,λ 2=ζ0 2,λ 3=−ζ0 2,(34) ξ1(c)=χ(c)+∂ ∂c[cχ(c)],ξ2(c)=−∂χ(c) ∂c, (35) ξ3(c)=−∂ ∂c[cχ(c)]. The eigenvalue λ2is d-fold degenerate. Normalization of the distribution function requires that the velocity distribution functions being considered be integrable. However, the strongest requirement is made now that they be elements of a Hilbert space with scalar product defined as g|h≡dcχ−1(c)g∗(c)h(c),(36) where g∗(c) is the complex conjugate of g(c). The operator (c) is not Hermitian and the eigenfunctions ξβ(c) are not orthogonal. Then, it is convenient to introduce a set of functions ξβ(c) biorthogonal to the above eigenfunctions, i.e., verifying ξβ|ξβ=δβ,β.(37) 041303-3
BREY, MAYNAR, AND GARC´ IA DE SORIA PHYSICAL REVIEW E 83, 041303 (2011) A convenient choice is [19,20] ξ1(c)=χ(c),ξ2(c)=cχ(c), ξ3(c)=c2 d+1 2χ(c). (38) Using the biorthogonal sets of functions, a projection operator Pover the hydrodynamic part of the Hilbert space can be defined by Pg(c)≡ d+2 β=1 ξβ(c)ξβ|g.(39) By means of P, the fluctuating one-particle distribution function can be decomposed into its hydrodynamic and nonhydrodynamic components, δ F(k,c,s)=Pδ F(k,c,s)+P⊥δ F(k,c,s),(40) where P⊥≡1−P. Application of P⊥to both sides of Eq. (14) and formal integration of the resulting equation yields [10] P⊥δ F(k,c,s)=U(k,c,s)P⊥δ F(k,c,0) +s 0 dsU(k,c,s)P⊥[−ik·cP ×δ F(k,c,s −s)+ S(k,c,s −s)],(41) where U(k,c,s)≡exp[sP⊥(k,c)P⊥].(42) The hypothesis is made now that for large enough s,the first term on the right-hand side of Eq. (41) becomes negligible. This is related with the ageing to hydrodynamics, implying that the nonhydrodynamic part of the initial condition is forgotten on the hydrodynamic time scale. This is a necessary condition for the existence of a hydrodynamic description and it has been extensively discussed in the context of the derivation of the Navier-Stokes equations for granular gases [21]. Let us notice that the validity of hydrodynamics for low-density systems of inelastic hard particles has been verified by means of numerical particle simulations in many different contexts and over a wide range of values of the inelasticity [22,23]. In addition, the limit of small wave vector kis considered and only terms up to first order in it are kept. Then, for large values of s,Eq.(41) becomes P⊥δ F(k,c,s)≃s 0 dsP⊥es(c)(−ik·c)Pδ F(k,c,s −s) +s 0 dsU(k,c,s)P⊥ S(k,c,s −s),(43) This expression will be used in the next sections to obtain explicit expression for the fluctuating fluxes and for the cooling rate in the Navier-Stokes approximation. IV. LANGEVIN EQUATION FOR THE VELOCITY FIELD By direct calculation, it is easily verified that dc(c)ξβ(c)=0.(44) Here ξβ(c) stands for any of the hydrodynamic eigenfunctions of (c)giveninEq.(35). Therefore, dc(c)Pδ F(k,c,s)=0 (45) and, consequently, δ Fcan be substituted by P⊥δ Fon the right-hand side of the expression for the fluctuating pressure tensor, Eq. (23). Using next Eq. (43)gives δ(k,s)≃δ(k,s) 2 I+δ1(k,s)+R(k,s),(46) where δ1(k,s)=s 0 dsdc(c)es(c) ×(−ik·c)Pδ F(k,c,s −s),(47) and R(k,s)=s 0 dsdc(c)U(k,c,s)P⊥ S(k,c,s −s). (48) Equation (21) implies that δω(k,s −s)≃e−sζ0/2δω(k,s) (49) to lowest order in k. Using this, it is obtained that [10] δ1ij (k,s)=−iη(s)kiδωj(k,s)+kjδωi(k,s) −2 dδij k·δω(k,s),(50) valid to first order in k. In the above expression, η(s)isthe time-dependent dimensionless shear viscosity of the HCS [24], η(s)≡1 d2+d−2 d i,j dcij (c)2,ij (c,s),(51) 2,ij (c,s)=s 0 dses(−ζ0/2)ξ2,i (c)cj.(52) Equation (51) agrees with the result found from the nonlinear Boltzmann equation by using the Chapman-Enskog algorithm, and its long time limit has been evaluated in the first Sonine approximation [22,25]. The last term on the right-hand side of Eq. (46) is defined in Eq. (48) and it has the property R(k,s)H=0,(53) that follows from Eq. (8). Moreover, in Appendix Bit is shown that for s1, s1, and to lowest order in k,itis Rij (k,s)Rlm(k,s)H = V2 Nδk,−kG(|s−s|)δilδjm +δimδjl −2 dδij δlm,(54) where G(s)≡1 d2+d−2 d i,j dc1 ×dc2ij (c1)ij (c2)es(c2) φH(c1,c2),(55) 041303-4
FLUCTUATING NAVIER-STOKES EQUATIONS FOR ... PHYSICAL REVIEW E 83, 041303 (2011) with φH(c1,c2) being the solution of the equation [(c1)+(c2)] φH(c1,c2)=−P(1) ⊥P(2) ⊥ (c1,c2).(56) The operators P(1) ⊥and P(2) ⊥are defined like P⊥, but acting on functions of the velocities c1and c2, respectively. An approximation will be introduced at this point. The noise amplitude (c1,c2) given by Eq. (11) has two contributions of a rather different physical origin. The first one reflects fluctuations induced by collisions as a consequence of two different particles colliding independently at the same position with the same environment. This contribution does not vanish, even for a molecular gas at equilibrium [26]. On the other hand, the second contribution to (c1,c2) is directly related with the velocity correlations between particles in the HCS and vanishes for an ordinary fluid at equilibrium. Here, the hypothesis will be made that only the hydrodynamic component of the velocity correlations, as extracted by the operator P(1)P(2), is relevant, while the remaining kinetic or nonhydrodynamic component is negligible. Some justifications for this assumption are provided in the discussion section of the paper. Then, the second term on the right-hand side of Eq. (11) is neglected when substituting it into Eq. (56) and the solution of this equation can be written down by simple inspection, φH(c1,c2)≃P(1) ⊥P(2) ⊥χ(c1)δ(c1−c2).(57) Use of Eqs. (46) and (50) into Eq. (21) gives the fluctuating Navier-Stokes equation for the velocity field, ∂ ∂s −ζ0 2δω(k,s)+ikδ(k,s) 2 +ηk2δω(k,s)+d−2 d k·δω(k,s) k= W(k,s),(58) where k≡k/k and the noise term W(k,s)=−ik·R(k,s) has the properties W(k,s)H=0,(59) Wi(k,s) Wj(k,s)H= V2 Nδk,−kG(|s−s|)k2 ×δij +d−2 d ki kj.(60) If Eq. (58) is particularized for the transversal component of the velocity field ω⊥≡ω− k·ω, the result reported in Ref. [10] is recovered. The quantity G(s) is evaluated approximately in Appendix C. The used approximation, which is an exact property for the Maxwell model for inelastic gases [27] reads +(c)xy(c)χ(c)≃λ4xy(c)χ(c).(61) In the above relation, +is the adjoint operator of and the eigenvalues λ4is determined self-consistently. The result is G(s)≃1+a2(α) 4esλ4,(62) with λ4=ζ0+4I(α) 1+a2(α).(63) The explicit form of the functions a2(α) and I(α)isgivenin Eqs. (A8) and (C6), respectively. In the elastic limit, α→1, the exponential above decays very fast on the hydrodynamic scale and it can be approximated by a δfunction. In this way, the classical result of the Landau-Lifschitz theory is recovered. It is interesting to evaluate the shear viscosity ηgiven in Eq. (51) in the same approximation as employed to calculate the function G(s), i.e., by using Eq. (61). In the long time limit s1 it is found in Appendix Cthat η≃8|I(α)| 1+a2(α)−ζ0−1 .(64) This result is very close to the one reported in Refs. [25] and [22], both being practically indistinguishable for α⩾0.6. Moreover, Eq. (64) coincides with the expression derived in Ref. [28] by using a modified Sonine expansion in which the Gaussian is replaced by the (approximated) distribution of the HCS, χ(c). V. LANGEVIN EQUATION FOR THE ENERGY FIELD First consider the term δζ(k,s)giveninEq.(27). It will be evaluated here by making an approximation in the same spirit as Eq. (61), namely by writing +(c)ξ3(c)≃λ3ξ3(c)=−ζ0 2ξ3(c).(65) Then, by realizing that Eq. (27) is equivalent to δζ(k,s)=−2ξ3|δ F,(66) it follows that δζ(k,s)≃ζ0 2[δ(k,s)+δρ(k,s)].(67) The same result is obtained if, instead of Eq. (65), the approximation ξ3|δ F≃ξ3|Pδ F(68) is employed. Therefore contributions to δζ(k,s)fromP⊥δ F are being neglected in Eq. (67). This includes, in particular, terms proportional to the gradients of the fluctuating hydrodynamic fields [see, for instance, Eq. (41)]. As for Eq. (61), the approximation given by Eq. (65) becomes an exact relation for the inelastic Maxwell model for granular gases [27]. Moreover, it has been shown that the linear transport coefficients associated with the gradient expansion of the average cooling rate are very small as compared with the similar contributions coming from the hydrodynamic fluxes [25]. Actually, the only contribution to the hydrodynamic equations from the average cooling rate that is kept in practically all the literature is the one of zeroth order in the gradients. Next, the heat flux term δφdefined in Eq. (24) has to be evaluated. It contains the function (c) that verifies a relationship similar to Eq. (44), dc(c)ξβ(c)=0,(69) 041303-5
BREY, MAYNAR, AND GARC´ IA DE SORIA PHYSICAL REVIEW E 83, 041303 (2011) for all the hydrodynamic modes ξβ(c). Therefore, Eq. (24)is equivalent to δφ(k,s)=dc(c)P⊥δ F(k,c,s),(70) and by means of Eq. (43), the energy flux can be decomposed as δφ(k,s)=δ1φ(k,s)+Z(k,s),(71) where δ1φ(k,s)=s 0 dsdc(c)es(c)(−ik·c)Pδ F(k,c,s −s) (72) and Z(k,s)=s 0 dsdc(c)U(k,c,s)P⊥ S(k,c,s −s). (73) The above expressions are valid up to first order in k. A direct calculation gives Pδ F(k,c,s)=δρ(k,s)ξ1(c)+ξ3(c) 2+δω(k,s)·ξ2(c) +1 2δ(k,s)ξ3(c),(74) and taking into account the isotropy of the operator (c), Eq. (72) can be rewritten as δ1φ(k,s)=s 0 dsdc(c)es(c)(−ik·c) ×δρ(k,s −s)ξ1(c)+1 2[δ(k,s −s) +δρ(k,s −s)]ξ3(c).(75) The next task is to evaluate the functions δρ(k,s −s) and δ(k,s −s) as functions of the fluctuating hydrodynamic fields at time sto lowest (zeroth) order in k, in order to have an expression for δ1φ(k,s) valid to first order in k.Useofthe balance hydrodynamic equations (20)–(22) leads to δρ(k,s −s)≃δρ(k,s),(76) δ(k,s −s)+δρ(k,s −s)≃eζ0s/2[δ(k,s)+δρ(k,s)] −e−ζ0(s−s)/2s s−s ds1eζ0s1/2 × S(k,s1).(77) When the above expressions are substituted into Eq. (75), two contributions that differ physically are identified. One of them is of hydrodynamic character and similar to the expression for the fluctuating heat flux for molecular gases derived by Landau and Lifshitz [9], while the other one is an intrinsic noise term following directly from the inelasticity of collisions, δ1φ(k,s)=δ(H) 1φ(k,s)+δ(I) 1φ(k,s).(78) After simple manipulations, the first contribution can be expressed as δ(H) 1φ(k,s)=s 0 dsdc(c)es(c)(−ik·c) ×δρ(k,s)ξ1(c)+1 2eζ0s/2[δρ(k,s) +δ(k,s)]ξ3(c)=−κikδ(k,s) −(μ−κ)ikδρ(k,s).(79) The right-hand side of this equation has the same form as the (generalized) Fourier law for dilute granular gases [22,25]. It involves two transport coefficients: the (thermal) heat conductivityκand the diffusive heat conductivity μ. Their expressions are: κ(s)=1 ddc(c)·3(c,s),(80) μ(s)=1 ddc(c)·1(c,s),(81) with 1(c,s)=s 0 dses(c)ξ1(c)c+23(c,s),(82) 3(c,s)=1 2s 0 dses[(c)+ζ0/2]ξ3(c)c.(83) As usual, κ(s) and μ(s) are expected to reach steady plateau values for large enough s, when the hydrodynamic description is accurate. Both transport coefficients have been evaluated in the first Sonine approximation [22,25]. The noise term in Eq. (78) is given by δ(I) 1φ(k,s)=1 2s 0 dsdc(c)es(c)ik·ce−ζ0(s−s)/2 ×s s−s ds1eζ0s1/2 S(k,s1)ξ3(c).(84) Substitution of Eqs. (67), (71), (78), and (79) into Eq. (22) yields ∂ ∂s +ζ0 2δ(k,s)+ζ0 2δρ(k,s)+id+2 d k·δω(k,s) +2 dk2[κδ(k,s)+(μ−κ)δρ(k,s)] = E(k,s),(85) where E(k,s)≡ S(k,s)−2i d k·Z(k,s)−2i d k·δ(I) 1φ(k,s) (86) is identified as the total noise term. The functions Z(k,s) and δ(I) 1φ(k,s) are defined in Eqs. (73) and (84), respectively, and the intrinsic inelastic noise term S(k,s)isgiveninEq.(28). It is trivial to verify that E(k,s)H=0,(87) while the corresponding correlation function is calculated in Appendices Dand Eunder well-defined and controlled 041303-6
FLUCTUATING NAVIER-STOKES EQUATIONS FOR ... PHYSICAL REVIEW E 83, 041303 (2011) approximations, which will be made explicit also below. The result reads E(k,s) E(k,s)H≃4 V2 Nζ0(α)a33(α)δ(s−s)δk,−k +(d+2) V2 Nd2δk,−kk21+(d+8) 2a2(α) +2dζ0(α)a33(α)[1 +2a2(α)] |λ5|−ζ0/2eλ5|s−s|, (88) valid for s,s1. In this expression a33(α) and λ5are given by Eqs. (D5) and (D13), respectively. The two main approximations made to derive the above expression are similar to those leading to Eqs. (60) and (62). First, the nonhydrodynamic component of particle velocity correlations is neglected once again. Second, it is written that +(c)(c)χ(c)≃λ5(c)χ(c),(89) and λ5is consistently obtained by means of Eq. (D12). In the elastic limit, the first term on the right-hand side of Eq. (88) vanishes, while the second one becomes proportional to a δ function of the time difference s−s. Moreover, the amplitude becomes proportional to the heat conductivity and the LandauLifschitz fluctuation-dissipation relation for the energy noise is recovered. When the expressions of thermal heat conductivityκand the diffusive heat conductivity μ,Eqs.(80) and (81) respectively, are evaluated using the two above approximations, the results read (see Appendix D) κ≃(d+2)[1 +2a2(α)] 2(2|λ5|−ζ0),(90) μ≃2κ−(d+2)[2 +a2(α)] 4|λ5|.(91) The above expressions hold in the limit of large time s. The above values for the thermal transport coefficients are indistinguishable of the results found in the first Sonine approximation [22,25] for all values of α. On the other hand, they are not equivalent to the expressions reported in Ref. [28], although both are very close for α0.65. Finally, to close the description provided by the fluctuating hydrodynamic equations derived here, the correlation between the fluctuating force appearing in the velocity equation and the fluctuating force in the energy equation is needed. It is verified in Appendix Fthat W(k,s) E(k,s)H=0,(92) as expected because of symmetry considerations. VI. FINAL REMARKS The objective here has been to derive fluctuating hydrodynamic equations for a dilute granular gas by extending methods which are familiar for normal gases. The main result obtained is the set of coupled Langevin-like equations (20), (58), and (85), describing the time evolution of the fluctuating hydrodynamic fields. These equations contain two kind of terms. There are terms which describe the hydrodynamic part of the fluxes and of the cooling rate. The former have the same form as the macroscopic hydrodynamic fluxes, involving the Navier-Stokes transport coefficients. They have been made explicit in the equations. On the other hand, there appear the nonhydrodynamic components of the fluxes as well as some additional terms in the equation for the energy due to the dissipation in collisions. These latter terms have zero average and have been combined all together to define the noise terms in the Langevin-like equations. The autocorrelation functions of the noise terms in the velocity and energy equations are given in Eqs. (60) and (88), respectively, while the cross correlation function between both noise terms is indicated to vanish in Eq. (92). Generalizing fluctuating hydrodynamics to granular fluids entails several important differences from normal molecular fluids.Primary among them are the following: (i) The homogeneous reference state about which fluctuations are considered is not the Boltzmann equilibrium state but the HCS. Its distribution function is not a simple function of the global invariants. Moreover, this reference state is time dependent, although it is possible to consider a stationary representation of it by using an appropriate time scale see [Eq. (2)]. (ii) Although the noise in the fluctuating inelastic Boltzmann equation is white, i.e., δcorrelated in time, the noise terms in the equations for the velocity and energy fields have finite relaxation times. Moreover, the amplitude of their correlation functions is not determined by the Navier-Stokes transport coefficients but involve new coefficients. In other words, the fluctuation-dissipation relations of the second kind [29] are not verified [30]. (iii) The noise term in the equation for the energy has a zeroth order in the gradients contribution. This noise term is intrinsic to the inelasticity of collisions and has no analog in molecular fluids. Of course, it vanishes in the elastic limit. The expressions for the two-time correlation functions of the noise terms have been computed using some approximations. It has been assumed that the velocity correlations between particles can be accurately approximated by their hydrodynamic part, identified as their projection onto the subspace of distributions generated by the hydrodynamic eigenfunctions of the linearized inelastic Boltzmann collision operator. The second approximation used consists in dealing with the hydrodynamic fluxes as if they were left eigenfunctions of the above-mentioned linear operator. The two hypothesis can be justified on the basis of the following features: (i) they have been shown to lead to some predictions that are in very good agreement with molecular dynamics simulation results. This includes the fluctuations of the total energy of the system [14,15] and also of the transversal component of the velocity field [10,31], (ii) if the approximations are applied to the formal expressions of the Navier-Stokes transport coefficients, very accurate expressions are obtained as discussed in previous sections of this paper, and (iii) the second approximation mentioned above is an exact property in the case of the inelastic Maxwell model kinetic equation. Some comments on the context and utility of the results in this work seem appropriate. The analysis has focused on the fluctuations of the hydrodynamic fields in the HCS. 041303-7
BREY, MAYNAR, AND GARC´ IA DE SORIA PHYSICAL REVIEW E 83, 041303 (2011) In this context, it seems interesting to investigate whether the long time tails of the correlation functions of the fluxes found in Ref. [32] are recovered within the theory developed here. On the other hand, in many experimental conditions of interest the system is far from a global homogeneous state. Nevertheless, the reference state studied here is relevant locally for more complex and realistic conditions, as it is the case of the equilibrium state in molecular fluids. For example, the transport coefficients such as the viscosity obtained here are the same functions of density and temperature as those in the associated nonlinear hydrodynamic equations applicable under more general conditions. Thus the context of relevance of the equations derived in the present work are expected to transcend the limitations associated with the state considered and extend to states for which the nonlinear Navier-Stokes are required to characterize the macroscopic hydrodynamic fields. Of course, more work is needed to verify this expectation, but some partial results already obtained for some particular nonhomogenous states of granular gases seem to suggest that this is the case. ACKNOWLEDGMENTS This research was supported by the Ministerio de Educaci´ on y Ciencia (Spain) through Grant No. FIS2008-01339 (partially financed by FEDER funds). APPENDIX A: DIMENSIONLESS LINEAR BOLTZMANN COLLISION OPERATOR The dimensionless binary collision operator T0(c1,c2)for inelastic hard spheres or disks is defined by T0(c1,c2)=d σ(c12 · σ)c12 · σα−2b−1 σ(c1,c2)−1, (A1) where c12 ≡c1−c2,d σis the solid angle element for the unit vector σ,is the Heaviside step function, and b−1 σ(c1,c2)is an operator changing all the functions of c1and c2to its right by the same functions of the precollisonal velocities c∗ 1and c∗ 2, given by c∗ 1≡b−1 σc1=c1−1+α 2α( σ·c12) σ, (A2) c∗ 2≡b−1 σc2=c2+1+α 2α( σ·c12) σ. The expression of the linearized Boltzmann collision operator (c)is[19] (c1)≡dc2T0(c1,c2)(1 +P12)χ(c2)−ζ0 2 ∂ ∂c1·c1. (A3) The operator P12 interchanges the labels of particles 1 and 2 of the quantities to its right, χ(c) is the scaled velocity distribution of the HCS defined in Eq. (12), and ζ0is the dimensionless cooling rate for the decay of the temperature of the HCS in the time scale s, dTH(s) ds =−ζ0TH(s),(A4) ζ0=(1 −α2)πd−1 2 2d+3 2ddc1dc2c3 12χ(c1)χ(c2).(A5) Approximated expressions for the distribution function of the HCS and for the cooling rate have been obtained by expanding the functions in Sonine polynomials and keeping only the lowest orders [5,33]. The distribution function has the form χ(c)=e−c2 πd/2[1 +a2(α)S(2)(c2)],(A6) where S(2)(c2)=c4 2−d+2 2c2+d(d+2) 8(A7) and a2(α)=16(1 −α)(1 −2α2) 9+24d+(8d−41)α+30α2−30α3.(A8) Equation (A6) is used all along this paper to carry out explicit calculations. The approximate expression for the cooling rate is: ζ0=√2π(d−1)/2(1 −α2) (d/2)d1+3a2(α) 16 .(A9) APPENDIX B: DERIVATION OF Eq. (54) From Eq. (48) and using Eq. (16), it follows that for small wave vector kit is R(k,s)R(k,s)H = V2 Nδk,−kdc1dc2(c1)(c2)s 0 ds1 ×s 0 ds2es1(c1)+s2(c2)δ(s−s−s1+s2) ×P(1) ⊥P(2) ⊥ (c1,c2),(B1) where it has been used that to lowest order in kthe inhomogeneous linear Boltzmann operator (k,c) can be replaced by the homogeneous one, (c). The projection operators P(1) ⊥ and P(2) ⊥act on functions of c1and c2, respectively. Suppose that s>s. The integration over s2can be easily carried out to get R(k,s)R(k,s)H = V2 Nδk,−kdc1dc2(c1)(c2)e(s−s)(c2) ×s 0 ds1es1[(c1)+(c2)]P(1) ⊥P(2) ⊥ (c1,c2).(B2) For s>s1 and assuming that all the nonhydrodynamic components of (c1,c2) correspond to negative eigenvalues of 041303-8
FLUCTUATING NAVIER-STOKES EQUATIONS FOR ... PHYSICAL REVIEW E 83, 041303 (2011) (c1) and (c2), the above relation can be simplified to R(k,s)R(k,s)H = V2 Nδk,−kdc1dc2(c1)(c2)e(s−s)(c2) φH(c1,c2), (B3) where φH(c1,c2) obeys Eq. (56). The symmetry of the tensor (c), the isotropy of the operator (c) and the invariance of (c1,c2) under rotations of c1and c2imply that the right-hand side of Eq. (B3) must have the form given in Eq. (54). The case s>s 1 follows trivially. APPENDIX C: APPROXIMATED EVALUATION OF THE FUNCTION G(s) DEFINED IN Eq. (55) Use of the approximation given in Eq. (57) into Eq. (55) gives G(s)≃1 d2+d−2 d i,j dcij (c)es(c)ij (c)χ(c),(C1) and taking into account the symmetry of the operator (c) and of the function χ(c), this expression is seen to be equivalent to G(s)=dcxy(c)es(c)ij (c)χ(c) =xyχ|esxyχ =es+xyχ|xyχ,(C2) where +(c) is the adjoint of (c) defined by g|h=+g|h∗,(C3) for arbitrary functions g(c) and h(c) of the Hilbert space. Now the approximation is made that xy(c)χ(c) is an eigenfunction of +, being λ4the eigenvalue, as expressed by Eq. (61). This approximation is prompted by the fact that it is exact for the inelastic Maxwell model of granular gases [27]. Use of Eq. (61) into Eq. (C2) yields G(s)=esλ4dcc2 xc2 yχ(c)=1+a2(α) 4esλ4,(C4) where the expression of χ(c) in the first Sonine approximation, Eq. (A6) has been employed to evaluate the velocity integral. To determine λ4,Eq.(61) is multiplied by xy(c) and integrated over cto get λ4=4 1+a2(α)dccxcy+(c)xy(c)χ(c).(C5) The evaluation of the velocity integral on the right-hand side of the above expression using once again Eq. (A6) is a lengthy but quite standard calculation. The result is given in Eq. (63), where I(α)=−(2d+3−3α)(1 +α)πd−1 2 2√2d(d+2)(d/2) 1+23a2(α) 16 .(C6) In the same approximation as introduced above, the expression for the shear viscosity, η(s), given by Eq. (51) in the limit of large s, becomes η≃∞ 0 dsdcxy(c)es[(c)−ζ0/2]ξ2,x (c)cy =∞ 0 dse−sζ0/2es+(χxy )|ξ2,x cy ≃χxy|ξ2,x(c)cy∞ 0 dse−s(ζ0/2−λ4) =−χxy |ξ2,x cyλ4−ζ0 2−1 .(C7) Taking into account that χxy|ξ2,xcy=1 2,(C8) Eq. (64) is obtained. APPENDIX D: CORRELATION OF THE NOISE TERM IN THE EQUATION FOR THE ENERGY FIELD In the following calculations, it will be assumed without loss of generality that s>s .FromEq.(86) and keeping only contributions up to order k2it is obtained that E(k,s) E(k,s)H ≃ S(k,s) S(k,s)H−4 d2k·Z(k,s)Z(k,s)H·k −2i d k·δ(I) 1φ(k,s) S(k,s)H−2i d k·Z(k,s) S(k,s)H. (D1) On writing the above equation, it has been taken into account that δ(I) 1φ(k,s)isatleastoffirstorderink, as it is directly realized from its expression in Eq. (84). Moreover, it has been used that S(k,s)δφ(k,s)H=0,(D2) as a consequence of Eq. (31). Consider first the self-correlation of the intrinsic noise S, given by Eq. (30) or, equivalently, by S(k,s) S(k,s)H=4 V2 Nδk,−kδ(s−s)dc ×dcξ3(c)ξ3(c) (c,c).(D3) The velocity integral appearing on the right-hand side of this equation can be evaluated exactly using the expression for χ(c) in the first Sonine approximation. Nevertheless, for the sake of consistency, here the nonhydrodynamic components of the particle velocity correlations in the HCS will be neglected, as it was done when solving Eq. (56). With this approximation, it was shown in Ref. [14] that S(k,s) S(k,s)H≃4 V2 Nζ0a33(α)δk,−kδ(s−s),(D4) 041303-9